On the Complexity of Parameterized Local Search for the Maximum Parsimony Problem

Authors Christian Komusiewicz , Simone Linz , Nils Morawietz , Jannik Schestag

Thumbnail PDF


  • Filesize: 0.84 MB
  • 18 pages

Document Identifiers

Author Details

Christian Komusiewicz
  • Institute of Computer Science, Friedrich Schiller Universität Jena, Germany
Simone Linz
  • School of Computer Science, University of Auckland, New Zealand
Nils Morawietz
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany
Jannik Schestag
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany

Cite AsGet BibTex

Christian Komusiewicz, Simone Linz, Nils Morawietz, and Jannik Schestag. On the Complexity of Parameterized Local Search for the Maximum Parsimony Problem. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Maximum Parsimony is the problem of computing a most parsimonious phylogenetic tree for a taxa set X from character data for X. A common strategy to attack this notoriously hard problem is to perform a local search over the phylogenetic tree space. Here, one is given a phylogenetic tree T and wants to find a more parsimonious tree in the neighborhood of T. We study the complexity of this problem when the neighborhood contains all trees within distance k for several classic distance functions. For the nearest neighbor interchange (NNI), subtree prune and regraft (SPR), tree bisection and reconnection (TBR), and edge contraction and refinement (ECR) distances, we show that, under the exponential time hypothesis, there are no algorithms with running time |I|^o(k) where |I| is the total input size. Hence, brute-force algorithms with running time |X|^𝒪(k) ⋅ |I| are essentially optimal. In contrast to the above distances, we observe that for the sECR-distance, where the contracted edges are constrained to form a subtree, a better solution within distance k can be found in k^𝒪(k) ⋅ |I|^𝒪(1) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Applied computing → Molecular evolution
  • Applied computing → Computational genomics
  • phylogenetic trees
  • parameterized complexity
  • tree distances
  • NNI
  • TBR


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Benjamin L. Allen and Mike Steel. Subtree transfer operations and their induced metrics on evolutionary trees. Ann. Comb., 5(1):1-15, 2001. Google Scholar
  2. Alexandre A. Andreatta and Celso C. Ribeiro. Heuristics for the phylogeny problem. J. Heuristics, 8(4):429-447, 2002. Google Scholar
  3. Hans L. Bodlaender, Michael R. Fellows, and Tandy J. Warnow. Two strikes against perfect phylogeny. In Proceedings of the 19th International Colloquium on Automata, Languages and Programming (ICALP '92), volume 623 of Lecture Notes in Computer Science, pages 273-283. Springer, 1992. Google Scholar
  4. Amir Carmel, Noa Musa-Lempel, Dekel Tsur, and Michal Ziv-Ukelson. The worst case complexity of maximum parsimony. J. Comput. Biol., 21(11):799-808, 2014. Google Scholar
  5. Jianer Chen, Benny Chor, Mike Fellows, Xiuzhen Huang, David W. Juedes, Iyad A. Kanj, and Ge Xia. Tight lower bounds for certain parameterized NP-hard problems. Inf. Comput., 201(2):216-231, 2005. URL: https://doi.org/10.1016/j.ic.2005.05.001.
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  7. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  8. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances A. Rosamond, Saket Saurabh, and Yngve Villanger. Local search: Is brute-force avoidable? J. Comput. Syst. Sci., 78(3):707-719, 2012. URL: https://doi.org/10.1016/j.jcss.2011.10.003.
  9. Joseph Felsenstein. Inferring Phylogenies. Sinauer Associates Sunderland, 2004. Google Scholar
  10. David Fernández-Baca and Jens Lagergren. A polynomial-time algorithm for near-perfect phylogeny. SIAM J. Comput., 32(5):1115-1127, 2003. URL: https://doi.org/10.1137/S0097539799350839.
  11. Walter M. Fitch. Toward defining the course of evolution: minimum change for a specific tree topology. Systematic Biology, 20(4):406-416, 1971. Google Scholar
  12. Les R. Foulds and Ronald L. Graham. The Steiner problem in phylogeny is NP-complete. Adv. Appl. Math., 3(1):43-49, 1982. Google Scholar
  13. Ganeshkumar Ganapathy, Vijaya Ramachandran, and Tandy Warnow. On contract-and-refine transformations between phylogenetic trees. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '04), pages 900-909, 2004. Google Scholar
  14. Ganeshkumar Ganapathy, Vijaya Ramachandran, and Tandy J. Warnow. Better hill-climbing searches for parsimony. In Proceedings of the 3rd International Workshop on Algorithms in Bioinformatics (WABI '03), volume 2812 of Lecture Notes in Computer Science, pages 245-258. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-39763-2_19.
  15. Serge Gaspers, Eun Jung Kim, Sebastian Ordyniak, Saket Saurabh, and Stefan Szeider. Don't be strict in local search! In Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI '12). AAAI Press, 2012. Google Scholar
  16. Adrien Goëffon, Jean-Michel Richer, and Jin-Kao Hao. Local search for the maximum parsimony problem. In Proceedings of the First International Conference on Advances in Natural Computation (ICNC '05), volume 3612 of Lecture Notes in Computer Science, pages 678-683. Springer, 2005. Google Scholar
  17. Adrien Goëffon, Jean-Michel Richer, and Jin-Kao Hao. Progressive tree neighborhood applied to the maximum parsimony problem. IEEE/ACM Trans. Comput. Biol. Bioinform., 5(1):136-145, 2008. URL: https://doi.org/10.1109/TCBB.2007.1065.
  18. Pablo A Goloboff. Character optimization and calculation of tree lengths. Cladistics, 9(4):433-436, 1993. Google Scholar
  19. Pablo A. Goloboff. Analyzing large data sets in reasonable times: Solutions for composite optima. Cladistics, 15(4):415-428, 1999. URL: https://doi.org/10.1006/clad.1999.0122.
  20. Maozu Guo, Jian-Fu Li, and Yang Liu. Improving the efficiency of p-ECR moves in evolutionary tree search methods based on maximum likelihood by neighbor joining. In Proceeding of the Second International Multi-Symposium of Computer and Computational Sciences (IMSCCS '07), pages 60-67. IEEE Computer Society, 2007. Google Scholar
  21. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. Google Scholar
  22. Richard M. Karp. Reducibility among combinatorial problems. In Proceedings of a Symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  23. Christian Komusiewicz and Nils Morawietz. Parameterized local search for vertex cover: When only the search radius is crucial. In Proceedings of the 17th International Symposium on Parameterized and Exact Computation (IPEC '22), volume 249 of LIPIcs, pages 20:1-20:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  24. Dániel Marx. Searching the k-change neighborhood for TSP is W[1]-hard. Oper. Res. Lett., 36(1):31-36, 2008. Google Scholar
  25. Kevin C. Nixon. The parsimony ratchet, a new method for rapid parsimony analysis. Cladistics, 15(4):407-414, 1999. URL: https://doi.org/10.1006/clad.1999.0121.
  26. Celso C. Ribeiro and Dalessandro Soares Vianna. A GRASP/VND heuristic for the phylogeny problem using a new neighborhood structure. Int. Trans. Oper. Res., 12(3):325-338, 2005. Google Scholar
  27. David F. Robinson. Comparison of labeled trees with valency three. J. Comb. Theory B, 11(2):105-119, 1971. Google Scholar
  28. David Sankoff. Minimal mutation trees of sequences. SIAM J. Appl. Math., 28(1):35-42, 1975. Google Scholar
  29. David Sankoff, Yvon Abel, and Jotun Hein. A treeperiodcentered a windowperiodcentered a hill; generalization of nearest-neighbor interchange in phylogenetic optimization. J. Classif., 11(2):209-232, 1994. Google Scholar
  30. Srinath Sridhar, Kedar Dhamdhere, Guy E. Blelloch, Eran Halperin, R. Ravi, and Russell Schwartz. Algorithms for efficient near-perfect phylogenetic tree reconstruction in theory and practice. IEEE/ACM Trans. Comput. Biol. Bioinform., 4(4):561-571, 2007. URL: https://doi.org/10.1109/TCBB.2007.1070.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail